10 125

10_124

10_126

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 125's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 125 at Knotilus!
	10_125 is also known as the pretzel knot P(5,-3,2).

Knot presentations

Planar diagram presentation	X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283
Gauss code	-1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code	4 8 14 2 -16 -18 6 -20 -10 -12
Conway Notation	[5,21,2-]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 1}, {6, 9}, {5, 7}, {4, 6}, {3, 5}, {11, 4}]

[edit Notes on presentations of 10 125]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 125"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X3849 X5,14,6,15 X20,16,1,15 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,20,13,19 X13,6,14,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 10, -2, 1, -3, 9, -10, 2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 -16 -18 6 -20 -10 -12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[5,21,2-]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(3,\{1,1,1,1,1,-2,-1,-1,-1,-2\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 10, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 8}, {7, 1}, {6, 9}, {5, 7}, {4, 6}, {3, 5}, {11, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 4.61196 A-Polynomial See Data:10 125/A-polynomial

[edit Notes for 10 125's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 125's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-2t^{2}+2t-1+2t^{-1}-2t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 11, 2 }
Jones polynomial	$-q^{4}+q^{3}-q^{2}+2q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-4a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}-3a^{2}-3a^{-2}+7$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+a^{3}z^{7}+2az^{7}+z^{7}a^{-1}-6a^{2}z^{6}-6z^{6}-6a^{3}z^{5}-11az^{5}-5z^{5}a^{-1}+11a^{2}z^{4}+2z^{4}a^{-2}+13z^{4}+10a^{3}z^{3}+17az^{3}+8z^{3}a^{-1}+z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-4a^{3}z-8az-6za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$
The A2 invariant	$-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+3+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}$
The G2 invariant	$q^{60}+q^{56}-q^{54}-q^{48}-2q^{44}-q^{40}-2q^{38}-q^{36}-q^{34}-2q^{32}-2q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2q^{14}+q^{12}+2q^{10}+2q^{8}+2q^{6}+2q^{4}+3q^{2}+1+2q^{-2}+2q^{-4}+q^{-6}+2q^{-8}+2q^{-10}+q^{-14}+2q^{-16}-q^{-18}+q^{-20}-q^{-24}+q^{-26}-q^{-28}-q^{-30}-q^{-34}-q^{-36}-q^{-38}-2q^{-40}-q^{-44}-q^{-46}-q^{-50}-q^{-56}+q^{-72}$

Further Quantum Invariants

Further quantum knot invariants for 10_125.

A1 Invariants.

Weight	Invariant
1	$-q^{9}+q^{3}+q+q^{-1}+q^{-3}-q^{-9}$
2	$q^{28}-q^{24}-q^{18}-q^{16}+q^{6}+q^{4}+q^{2}+2+q^{-2}+q^{-6}-q^{-12}-q^{-18}-q^{-20}+q^{-24}$
3	$-q^{57}+q^{53}+q^{51}-q^{47}+q^{43}+q^{41}-q^{37}-q^{35}-q^{27}-q^{25}-q^{23}-q^{17}-q^{15}+q^{13}+2q^{11}+2q^{9}+q^{5}+2q^{3}+q+q^{-3}+q^{-5}+q^{-13}-q^{-17}-2q^{-19}-q^{-21}+q^{-23}+2q^{-25}-q^{-27}-2q^{-29}-q^{-31}+2q^{-33}-q^{-37}+q^{-41}+q^{-43}-q^{-45}$
5	$-q^{145}+q^{141}+q^{139}+q^{137}-q^{133}-2q^{131}-q^{129}+q^{125}+2q^{123}+2q^{121}-2q^{117}-2q^{115}-2q^{113}-q^{111}+q^{109}+2q^{107}+2q^{105}+q^{103}-2q^{99}-2q^{97}-q^{95}+q^{91}+2q^{89}+2q^{87}+q^{85}-2q^{81}-2q^{79}-q^{77}+q^{75}+4q^{73}+4q^{71}+2q^{69}-q^{67}-4q^{65}-5q^{63}-2q^{61}+2q^{59}+4q^{57}+3q^{55}-2q^{53}-5q^{51}-6q^{49}-3q^{47}+2q^{45}+4q^{43}+3q^{41}-4q^{37}-4q^{35}-2q^{33}+2q^{31}+4q^{29}+4q^{27}+q^{25}-2q^{23}-3q^{21}+3q^{17}+5q^{15}+3q^{13}-q^{11}-3q^{9}-2q^{7}+2q^{5}+4q^{3}+3q-3q^{-3}-2q^{-5}+q^{-7}+3q^{-9}+2q^{-11}-q^{-13}-q^{-15}+q^{-19}-2q^{-23}-3q^{-25}-q^{-27}+q^{-29}+3q^{-31}+2q^{-33}+q^{-35}-q^{-37}-3q^{-39}-3q^{-41}-q^{-43}+2q^{-45}+5q^{-47}+5q^{-49}+q^{-51}-5q^{-53}-7q^{-55}-5q^{-57}+q^{-59}+6q^{-61}+8q^{-63}+2q^{-65}-5q^{-67}-7q^{-69}-5q^{-71}+q^{-73}+4q^{-75}+3q^{-77}+q^{-79}-q^{-81}-q^{-83}+q^{-89}+q^{-91}+2q^{-93}-q^{-97}-2q^{-99}-q^{-101}+q^{-107}$
6	$q^{204}-q^{200}-q^{198}-q^{196}+2q^{190}+2q^{188}+q^{186}-q^{182}-2q^{180}-3q^{178}-q^{176}+2q^{172}+3q^{170}+3q^{168}+2q^{166}-q^{164}-2q^{162}-3q^{160}-3q^{158}-2q^{156}+2q^{152}+3q^{150}+3q^{148}+2q^{146}-2q^{142}-3q^{140}-3q^{138}-2q^{136}-q^{134}+q^{132}+2q^{130}+4q^{128}+3q^{126}+q^{124}-q^{122}-4q^{120}-5q^{118}-5q^{116}-q^{114}+3q^{112}+6q^{110}+7q^{108}+6q^{106}+q^{104}-5q^{102}-7q^{100}-7q^{98}-2q^{96}+4q^{94}+9q^{92}+9q^{90}+5q^{88}-q^{86}-8q^{84}-10q^{82}-7q^{80}-q^{78}+4q^{76}+8q^{74}+8q^{72}+2q^{70}-4q^{68}-8q^{66}-9q^{64}-7q^{62}+6q^{58}+8q^{56}+5q^{54}-6q^{50}-10q^{48}-7q^{46}-q^{44}+7q^{42}+10q^{40}+9q^{38}+2q^{36}-6q^{34}-9q^{32}-6q^{30}+q^{28}+7q^{26}+11q^{24}+6q^{22}-2q^{20}-7q^{18}-8q^{16}-2q^{14}+4q^{12}+9q^{10}+6q^{8}-q^{6}-5q^{4}-6q^{2}-1+4q^{-2}+7q^{-4}+5q^{-6}-q^{-8}-4q^{-10}-5q^{-12}-2q^{-14}+q^{-16}+4q^{-18}+2q^{-20}-q^{-22}-2q^{-24}-q^{-26}+2q^{-28}+q^{-30}-2q^{-34}-3q^{-36}-q^{-38}+q^{-40}+5q^{-42}+5q^{-44}+3q^{-46}-q^{-48}-4q^{-50}-5q^{-52}-6q^{-54}-3q^{-56}+2q^{-58}+7q^{-60}+9q^{-62}+8q^{-64}+q^{-66}-7q^{-68}-14q^{-70}-13q^{-72}-6q^{-74}+5q^{-76}+16q^{-78}+15q^{-80}+8q^{-82}-5q^{-84}-15q^{-86}-18q^{-88}-9q^{-90}+6q^{-92}+14q^{-94}+14q^{-96}+5q^{-98}-4q^{-100}-12q^{-102}-9q^{-104}+6q^{-108}+8q^{-110}+5q^{-112}+q^{-114}-4q^{-116}-5q^{-118}-2q^{-120}+q^{-122}+2q^{-124}+2q^{-126}+q^{-128}-2q^{-130}-4q^{-132}-2q^{-134}+q^{-138}+2q^{-140}+2q^{-142}+q^{-144}-q^{-146}-q^{-148}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{12}-q^{10}-q^{8}+q^{4}+2q^{2}+3+2q^{-2}+q^{-4}-q^{-8}-q^{-10}-q^{-12}$
1,1	$q^{36}+2q^{32}-2q^{30}+2q^{28}-2q^{26}+2q^{24}-2q^{22}-4q^{20}-2q^{18}-6q^{16}-5q^{12}+2q^{10}+6q^{6}+5q^{4}+8q^{2}+8+6q^{-2}+5q^{-4}-2q^{-6}-2q^{-8}-4q^{-10}-5q^{-12}-2q^{-14}-2q^{-16}+2q^{-20}-2q^{-26}+q^{-36}$
2,0	$q^{34}+q^{32}+q^{30}-q^{24}-2q^{22}-3q^{20}-3q^{18}-2q^{16}-2q^{14}-2q^{12}-q^{10}+q^{8}+3q^{6}+5q^{4}+6q^{2}+8+5q^{-2}+4q^{-4}-q^{-8}-3q^{-10}-2q^{-12}-3q^{-14}-3q^{-16}-2q^{-18}-q^{-20}-q^{-24}+q^{-26}+q^{-28}+q^{-30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{26}+q^{22}-q^{18}-2q^{16}-3q^{14}-4q^{12}-5q^{10}-2q^{8}+5q^{4}+7q^{2}+10+8q^{-2}+5q^{-4}+q^{-6}-2q^{-8}-4q^{-10}-4q^{-12}-3q^{-14}-2q^{-16}-q^{-18}+q^{-30}$
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{15}-q^{13}-2 q^{11}-q^9-q^7+q^5+3 q^3+4 q+4 q^{-1} +3 q^{-3} + q^{-5} - q^{-7} - q^{-9} -2 q^{-11} - q^{-13} - q^{-15} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}+2 q^{40}+q^{36}+q^{32}+q^{30}+q^{26}-4 q^{24}-2 q^{22}-9 q^{20}-7 q^{18}-14 q^{16}-11 q^{14}-11 q^{12}-5 q^{10}+5 q^8+11 q^6+24 q^4+22 q^2+30+18 q^{-2} +16 q^{-4} +3 q^{-6} -6 q^{-8} -11 q^{-10} -17 q^{-12} -13 q^{-14} -15 q^{-16} -5 q^{-18} -4 q^{-20} +4 q^{-22} +3 q^{-24} +5 q^{-26} +3 q^{-28} - q^{-30} -2 q^{-34} + q^{-48} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}+q^{30}+2 q^{28}+2 q^{26}+2 q^{24}-3 q^{20}-6 q^{18}-9 q^{16}-11 q^{14}-12 q^{12}-8 q^{10}-3 q^8+5 q^6+11 q^4+18 q^2+21+19 q^{-2} +14 q^{-4} +8 q^{-6} -6 q^{-10} -9 q^{-12} -10 q^{-14} -10 q^{-16} -7 q^{-18} -4 q^{-20} -2 q^{-22} - q^{-24} + q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} + q^{-36} }
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{18}-q^{16}-2 q^{14}-2 q^{12}-2 q^{10}-q^8+q^6+3 q^4+5 q^2+5+5 q^{-2} +3 q^{-4} + q^{-6} - q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} - q^{-16} - q^{-18} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{26}-q^{22}-q^{18}-q^{14}+q^{10}+2 q^6+q^4+3 q^2+2 q^{-2} + q^{-4} + q^{-6} - q^{-14} - q^{-18} - q^{-30} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{44}+q^{36}-q^{32}-q^{30}-q^{26}-2 q^{24}-2 q^{22}-q^{20}-q^{18}-2 q^{16}-q^{14}-q^{12}+q^{10}+q^8+3 q^6+2 q^4+4 q^2+4+4 q^{-2} +2 q^{-4} +3 q^{-6} +2 q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-16} - q^{-18} -2 q^{-20} -2 q^{-22} - q^{-24} - q^{-26} - q^{-28} - q^{-30} + q^{-48} }

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{34}+q^{30}+q^{26}-q^{24}-q^{22}-3q^{20}-4q^{18}-5q^{16}-6q^{14}-4q^{12}-3q^{10}+q^{8}+3q^{6}+8q^{4}+9q^{2}+12+9q^{-2}+8q^{-4}+4q^{-6}+q^{-8}-2q^{-10}-4q^{-12}-5q^{-14}-5q^{-16}-3q^{-18}-3q^{-20}-q^{-22}-q^{-24}+q^{-42}$

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}+q^{56}-q^{54}-q^{48}-2 q^{44}-q^{40}-2 q^{38}-q^{36}-q^{34}-2 q^{32}-2 q^{28}-q^{26}+q^{24}-q^{22}+q^{20}+q^{16}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^6+2 q^4+3 q^2+1+2 q^{-2} +2 q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-14} +2 q^{-16} - q^{-18} + q^{-20} - q^{-24} + q^{-26} - q^{-28} - q^{-30} - q^{-34} - q^{-36} - q^{-38} -2 q^{-40} - q^{-44} - q^{-46} - q^{-50} - q^{-56} + q^{-72} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 125"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{3}-2t^{2}+2t-1+2t^{-1}-2t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+4z^{4}+3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 11, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{4}+q^{3}-q^{2}+2q-1+2q^{-1}-q^{-2}+q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-4a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}-3a^{2}-3a^{-2}+7$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{2}z^{8}+z^{8}+a^{3}z^{7}+2az^{7}+z^{7}a^{-1}-6a^{2}z^{6}-6z^{6}-6a^{3}z^{5}-11az^{5}-5z^{5}a^{-1}+11a^{2}z^{4}+2z^{4}a^{-2}+13z^{4}+10a^{3}z^{3}+17az^{3}+8z^{3}a^{-1}+z^{3}a^{-3}-8a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-15z^{2}-4a^{3}z-8az-6za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 125"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-2 t^2+2 t-1+2 t^{-1} -2 t^{-2} + t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^4+q^3-q^2+2 q-1+2 q^{-1} - q^{-2} + q^{-3} - q^{-4} } }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(3, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 78} $2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -64} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 936} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} ${\frac {9151}{10}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2374}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{474}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{75}{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{289}{10}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 125. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 χ 9                 1 -1 7                   0 5             1 1   0 3           1       1 1         1 2       1 -1       1           1 -3       1           1 -5   1 1             0 -7                   0 -9 1                 -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} $i=3$ $r=-5$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{11}-q^{10}-q^9+q^8-q^6+q^4-q^3+q^2+2 q^{-1} - q^{-2} +2 q^{-4} -2 q^{-5} +2 q^{-7} -2 q^{-8} - q^{-9} +2 q^{-10} - q^{-11} - q^{-12} + q^{-13} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{21}+2 q^{20}-q^{18}-2 q^{17}+3 q^{16}+2 q^{15}-4 q^{14}-3 q^{13}+4 q^{12}+5 q^{11}-5 q^{10}-5 q^9+3 q^8+6 q^7-4 q^6-4 q^5+2 q^4+6 q^3-4 q^2-3 q+2+5 q^{-1} -3 q^{-2} -2 q^{-3} + q^{-4} +4 q^{-5} - q^{-6} -2 q^{-7} +2 q^{-9} - q^{-10} - q^{-11} + q^{-13} - q^{-14} - q^{-15} + q^{-16} + q^{-17} - q^{-18} -2 q^{-19} + q^{-20} +2 q^{-21} -2 q^{-23} + q^{-25} + q^{-26} - q^{-27} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+q^{32}+q^{31}-q^{29}-2 q^{28}+3 q^{27}+q^{26}-q^{25}-4 q^{24}-q^{23}+7 q^{22}+2 q^{21}-3 q^{20}-8 q^{19}-q^{18}+10 q^{17}+4 q^{16}-2 q^{15}-11 q^{14}-3 q^{13}+11 q^{12}+4 q^{11}-q^{10}-10 q^9-3 q^8+9 q^7+3 q^6-8 q^4-3 q^3+8 q^2+3 q+1-7 q^{-1} -4 q^{-2} +6 q^{-3} +4 q^{-4} +2 q^{-5} -5 q^{-6} -5 q^{-7} +4 q^{-8} +3 q^{-9} +3 q^{-10} -2 q^{-11} -5 q^{-12} + q^{-13} + q^{-14} +3 q^{-15} + q^{-16} -4 q^{-17} - q^{-18} - q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} - q^{-24} - q^{-25} +3 q^{-26} -2 q^{-27} + q^{-28} - q^{-30} +3 q^{-31} -3 q^{-32} +4 q^{-36} -2 q^{-37} - q^{-38} - q^{-39} - q^{-40} +3 q^{-41} - q^{-44} - q^{-45} + q^{-46} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{51}-q^{50}-q^{48}-q^{47}+q^{46}+2 q^{45}+q^{44}-q^{43}-q^{42}-2 q^{41}+q^{40}+q^{39}+q^{38}+q^{37}+q^{36}-q^{35}-2 q^{34}-5 q^{33}-q^{32}+3 q^{31}+8 q^{30}+5 q^{29}-4 q^{28}-10 q^{27}-7 q^{26}+q^{25}+10 q^{24}+11 q^{23}-10 q^{21}-10 q^{20}-2 q^{19}+8 q^{18}+11 q^{17}+2 q^{16}-8 q^{15}-9 q^{14}-q^{13}+6 q^{12}+9 q^{11}-7 q^9-7 q^8+5 q^6+8 q^5-4 q^3-6 q^2-2 q+2+7 q^{-1} +3 q^{-2} - q^{-3} -5 q^{-4} -4 q^{-5} -2 q^{-6} +6 q^{-7} +5 q^{-8} +3 q^{-9} -3 q^{-10} -6 q^{-11} -5 q^{-12} +3 q^{-13} +6 q^{-14} +6 q^{-15} -6 q^{-17} -7 q^{-18} - q^{-19} +4 q^{-20} +6 q^{-21} +4 q^{-22} -3 q^{-23} -6 q^{-24} -3 q^{-25} - q^{-26} +3 q^{-27} +5 q^{-28} - q^{-30} -2 q^{-31} -3 q^{-32} - q^{-33} +2 q^{-34} + q^{-35} +2 q^{-36} + q^{-37} - q^{-38} - q^{-39} - q^{-40} - q^{-41} + q^{-42} + q^{-43} + q^{-44} - q^{-47} - q^{-51} + q^{-53} + q^{-54} + q^{-55} -2 q^{-57} -2 q^{-58} + q^{-60} + q^{-61} +2 q^{-62} -2 q^{-64} - q^{-65} + q^{-68} + q^{-69} - q^{-70} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{71}+2 q^{69}+q^{68}-q^{66}-q^{65}-3 q^{64}-2 q^{63}+4 q^{62}+4 q^{61}+q^{60}-q^{59}-2 q^{58}-6 q^{57}-5 q^{56}+5 q^{55}+9 q^{54}+5 q^{53}+2 q^{52}-4 q^{51}-12 q^{50}-14 q^{49}+2 q^{48}+17 q^{47}+14 q^{46}+11 q^{45}-4 q^{44}-20 q^{43}-29 q^{42}-7 q^{41}+20 q^{40}+24 q^{39}+24 q^{38}+3 q^{37}-19 q^{36}-40 q^{35}-18 q^{34}+13 q^{33}+23 q^{32}+31 q^{31}+11 q^{30}-12 q^{29}-39 q^{28}-20 q^{27}+8 q^{26}+18 q^{25}+28 q^{24}+12 q^{23}-11 q^{22}-36 q^{21}-16 q^{20}+10 q^{19}+18 q^{18}+24 q^{17}+10 q^{16}-13 q^{15}-35 q^{14}-14 q^{13}+11 q^{12}+19 q^{11}+21 q^{10}+9 q^9-12 q^8-32 q^7-12 q^6+8 q^5+16 q^4+18 q^3+10 q^2-9 q-26-10 q^{-1} +5 q^{-2} +11 q^{-3} +13 q^{-4} +11 q^{-5} -5 q^{-6} -19 q^{-7} -7 q^{-8} +5 q^{-10} +7 q^{-11} +12 q^{-12} -11 q^{-14} -2 q^{-15} -4 q^{-16} - q^{-17} +9 q^{-19} +3 q^{-20} -3 q^{-21} +5 q^{-22} -3 q^{-23} -4 q^{-24} -8 q^{-25} +3 q^{-26} + q^{-28} +11 q^{-29} +2 q^{-30} - q^{-31} -10 q^{-32} -3 q^{-33} -7 q^{-34} - q^{-35} +12 q^{-36} +6 q^{-37} +5 q^{-38} -4 q^{-39} -3 q^{-40} -11 q^{-41} -6 q^{-42} +6 q^{-43} +3 q^{-44} +7 q^{-45} +3 q^{-46} +3 q^{-47} -7 q^{-48} -6 q^{-49} + q^{-50} -3 q^{-51} +2 q^{-52} +3 q^{-53} +5 q^{-54} - q^{-55} - q^{-56} +2 q^{-57} -4 q^{-58} - q^{-59} - q^{-60} + q^{-61} - q^{-62} +5 q^{-64} -2 q^{-65} + q^{-66} - q^{-68} -2 q^{-69} -2 q^{-70} +4 q^{-71} -3 q^{-72} + q^{-73} + q^{-74} + q^{-75} - q^{-77} +4 q^{-78} -4 q^{-79} - q^{-80} - q^{-81} +5 q^{-85} - q^{-86} - q^{-88} - q^{-89} -2 q^{-90} - q^{-91} +3 q^{-92} + q^{-94} - q^{-97} - q^{-98} + q^{-99} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

10_124 10_126